Distant future of the Earth and Sun Revisited

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					                                           Mon. Not. R. Astron. Soc. 000, 1–10 (2008)       Printed 3 February 2008      (MN L TEX style file v2.2)
                                                                                                                             A




                                           Distant future of the Sun and Earth revisited

                                           K.-P. Schr¨der1⋆ and Robert Connon Smith2†
                                           1
                                                     o
                                                                        ıa,
                                               Departamento de Astronom´ Universidad de Guanajuato, A.P. 144, Guanajuato, C.P. 36000, GTO, M´xicoe
                                           2 Astronomy   Centre, Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, UK
arXiv:0801.4031v1 [astro-ph] 25 Jan 2008




                                           Accepted 2008 ....; Received 200 ....; in original form 2007 September 25



                                                                              ABSTRACT
                                                                              We revisit the distant future of the Sun and the solar system, based on stellar models
                                                                              computed with a thoroughly tested evolution code. For the solar giant stages, mass-
                                                                              loss by the cool (but not dust-driven) wind is considered in detail. Using the new and
                                                                                                                        o
                                                                              well-calibrated mass-loss formula of Schr¨der & Cuntz (2005, 2007), we find that the
                                                                              mass lost by the Sun as an RGB giant (0.332 M⊙ , 7.59 Gy from now) potentially gives
                                                                              planet Earth a significant orbital expansion, inversely proportional to the remaining
                                                                              solar mass.
                                                                                   According to these solar evolution models, the closest encounter of planet Earth
                                                                              with the solar cool giant photosphere will occur during the tip-RGB phase. During
                                                                              this critical episode, for each time-step of the evolution model, we consider the loss
                                                                              of orbital angular momentum suffered by planet Earth from tidal interaction with
                                                                              the giant Sun, as well as dynamical drag in the lower chromosphere. As a result of
                                                                              this, we find that planet Earth will not be able to escape engulfment, despite the
                                                                              positive effect of solar mass-loss. In order to survive the solar tip-RGB phase, any
                                                                              hypothetical planet would require a present-day minimum orbital radius of about 1.15
                                                                              AU. The latter result may help to estimate the chances of finding planets around
                                                                              White Dwarfs.
                                                                                   Furthermore, our solar evolution models with detailed mass-loss description pre-
                                                                              dict that the resulting tip-AGB giant will not reach its tip-RGB size. Compared to
                                                                              other solar evolution models, the main reason is the more significant amount of mass
                                                                              lost already in the RGB phase of the Sun. Hence, the tip-AGB luminosity will come
                                                                              short of driving a final, dust-driven superwind, and there will be no regular solar plan-
                                                                              etary nebula (PN). The tip-AGB is marked by a last thermal pulse and the final mass
                                                                              loss of the giant may produce a circumstellar (CS) shell similar to, but rather smaller
                                                                              than, that of the peculiar PN IC 2149 with an estimated total CS shell mass of just a
                                                                              few hundredths of a solar mass.
                                                                              Key words: Sun: evolution – Sun: solar-terrestrial relations – stars: supergiants –
                                                                              stars: mass loss – stars: evolution – stars: white dwarfs



                                           1     INTRODUCTION                                                     Apps 2001 (hereafter SSA)), and has been discussed very
                                                                                                                  recently by Laughlin (2007).
                                           Climate change and global warming may have drastic effects
                                           on the human race in the near future, over human time-                      Theoretical models of solar evolution tell us that the
                                           scales of decades or centuries. However, it is also of interest,       Sun started on the zero-age main sequence (ZAMS) with
                                           and of relevance to the far future of all living species, to           a luminosity only about 70% of its current value, and it
                                           consider the much longer-term effects of the gradual heating            has been a long-standing puzzle that the Earth seems none
                                           of the Earth by a more luminous Sun as it evolves towards              the less to have maintained a roughly constant temperature
                                           its final stage as a white dwarf star. This topic has been              over its life-time, in contrast to what an atmosphere-free
                                           explored on several occasions (e.g. Sackmann, Boothroyd &              model of irradiation would predict. Part of the explanation
                                                                                          o
                                           Kraemer 1993, Rybicki & Denis 2001, Schr¨der, Smith &                  may be that the early atmosphere, rich in CO2 that was
                                                                                                                  subsequently locked up in carbonates, kept the temperature
                                                                                                                  up by a greenhouse effect which decreased in effectiveness
                                                                                                                  at just the right rate to compensate for the increasing solar
                                           ⋆ E-mail: kps@astro.ugto.mx (KPS)                                                  o
                                                                                                                  flux. The rˆle of clouds, and their interaction with galactic
                                           † E-mail: R.C.Smith@sussex.ac.uk (RCS)                                 cosmic rays (CR), may also be important: there is now some
2               o
      K.-P. Schr¨der and R.C. Smith
evidence (Svensmark 2007; but see Harrison et al. 2007 and        has since been further improved and calibrated rather care-
Priest et al. 2007) that cosmic rays encourage cloud cover        fully against observation, so that we believe that it is cur-
at low altitudes, so that a higher CR flux would lead to a         rently the best available representation of mass loss from
higher albedo and lower surface temperature. The stronger                                              o
                                                                  stars with non-dusty winds (Schr¨der & Cuntz 2005, 2007
solar wind from the young Sun would have excluded galactic        – see Section 2, where we explore the consequences of this
cosmic rays, so cloud cover on the early Earth may have been      improved mass-loss formulation).
less than now, allowing the full effect of the solar flux to be          However, although we have considerably reduced the
felt.                                                             uncertainties in the mass-loss rate, there is another factor
      What of the future? Although the Earth’s atmosphere         that works against the favourable effects of mass loss: tidal
may not be able to respond adequately on a short time-            interactions. Expansion of the Sun will cause it to slow its
scale to the increased greenhouse effect of carbon dioxide         rotation, and even simple conservation of angular momen-
and methane released into the atmosphere by human activ-          tum predicts that by the time the radius has reached some
ity, there is still the possibility, represented by James Love-   250 times its present value (cf. Table 1) the rotation period
lock’s Gaia hypothesis (Lovelock 1979, 1988, 2006), that the      of the Sun will have increased to several thousand years in-
biosphere may on a longer time-scale be able to adjust it-        stead of its present value of under a month; effects of mag-
self to maintain life. Some doubt has been cast on that view      netic braking will lengthen this period even more. This is so
by recent calculations (Scaife, private communication, 2007;      much longer than the orbital period of the Earth, even in
for details, see e.g. Cox et al. 2004, Betts et al. 2004) which   its expanded orbit, that the tidal bulge raised on the Sun’s
suggest that, on the century timescale, the inclusion of bio-     surface by the Earth will pull the Earth back in its orbit,
spheric processes in climate models actually leads to an in-      causing it to spiral inwards.
crease in carbon dioxide emissions, partly through a feed-             This effect was considered by Rybicki & Denis (2001),
back that starts to dominate as vegetation dies back. In any      who argued that Venus was probably engulfed, but that the
case, it is clear that the time will come when the increasing     Earth might survive. An earlier paper by Rasio et al. (1996)
solar flux will raise the mean temperature of the Earth to a       also considered tidal effects and concluded on the contrary
level that not even biological or other feedback mechanisms       that the Earth would probably be engulfed. However, the
can prevent. There will certainly be a point at which life is     Rybicki & Denis calculations were based on combining an-
no longer sustainable, and we shall discuss this further in       alytic representations of evolution models (of Hurley, Pols
Section 3.                                                        & Tout 2000) with the original Reimers’ mass-loss formula
      After that, the fate of the Earth is of interest mainly     rather than on full solar evolution calculations with a well-
insofar as it tells us what we might expect to see in systems     calibrated mass-loss formulation. The Rasio et al. paper also
that we observe now at a more advanced stage of evolution.        employed the original Reimers’ formula, and both papers use
We expect the Sun to end up as a white dwarf – do we expect       somewhat different treatments of tidal drag. We have there-
there to be any planets around it, and in particular do we        fore re-considered this problem in detail, with our own evolu-
expect any small rocky planets like the Earth?                    tionary calculations and an improved mass-loss description
      The question of whether the Earth survives has proved       as the basis; full details are given in Sections 2 and 4.
somewhat tricky to determine, with some authors arguing
that the Earth survives (e.g. SSA) and others (e.g. Sack-
mann et al. 1993) claiming that even Venus survives, while
                                                                  2   SOLAR EVOLUTION MODEL WITH MASS
general textbooks (e.g. Prialnik 2000, p.10) tend to say that
                                                                      LOSS
the Earth is engulfed. A simple model (e.g. SSA), ignoring
mass loss from the Sun, shows clearly that all the planets        In order to describe the long-term solar evolution, we use the
out to and including Mars are engulfed, either at the red         Eggleton evolution code (Eggleton 1971, 1972, 1973) in the
giant branch (RGB) phase – Mercury and Venus – or at the          version described by Pols et al. (1995, 1998), which has up-
later asymptotic giant branch (AGB) phase – the Earth and         dated opacities and an improved equation of state. Among
Mars. However, the Sun loses a significant amount of mass          other desirable characteristics, his code uses a self-adapting
during its giant branch evolution, and that has the effect         mesh and a ∇-based prescription of “overshooting”, which
that the planetary orbits expand, and some of them keep           has been well-tested and calibrated with giant stars in eclips-
ahead of the advancing solar photosphere. The effect is en-                                           o
                                                                  ing binaries (for details, see Schr¨der et al. 1997, Pols et al.
hanced by the fact (SSA) that when mass loss is included                      o
                                                                  1997, Schr¨der 1998). Because of the low mass and a non-
the solar radius at the tip of the AGB is comparable to that      convective core, solar evolution models are, however, not
at the tip of the RGB, instead of being much larger; Mars         subject to any MS (main sequence) core-overshooting. In
certainly survives, and it appears (SSA) that the Earth does      use, the code is very fast, and mass-loss is accepted simply
also.                                                             as an outer boundary condition.
      The crucial question here is: what is the rate of mass            As already pointed out by VandenBerg (1991), evolu-
loss in real stars? Ultimately this must be determined from       tion codes have the tendency to produce, with their most
observations, but in practice these must be represented           evolved models, effective temperatures that are slightly
by some empirical formula. Most people use the classical          higher than the empirically determined values. The reason
Reimers’ formula (Reimers 1975, 1977), but there is consid-       lies, probably, in an inadequacy of both low-temperature
erable uncertainty in the value to be used for his parameter      opacities and mixing-length theory at low gravity. With the
η, and different values are needed to reproduce the observa-       latter, we should expect a reduced efficiency of the con-
tions in different parameter regimes. In our own calculations      vective energy transport for very low gravity, because the
(SSA) we used a modification of the Reimers’ formula, which        largest eddies are cut out once the ratio of eddy-size to stellar
                                                                               Distant future of Sun and Earth                  3
radius has increased too much with g −1 . Hence, as described     Table 1. Main physical properties of characteristic solar models
        o
by Schr¨der, Winters & Sedlmayr (1999), our mixing-length
parameter, normally α = 2.0 for log g < 1.94, receives a           Phase           Age/Gy       L/L⊙      Teff /K   R/R⊙   MSun /M⊙
small adjustment in the form of a gradual reduction for su-
pergiant models, reaching α = 1.67 at log g = 0.0. With            ZAMS              0.00        0.70      5596    0.89      1.000
this economical adjustment, our evolution models now give          present           4.58        1.00      5774    1.00      1.000
                                                                   MS:hottest        7.13        1.26      5820    1.11      1.000
a better match to empirically determined effective tempera-
                                                                   MS:final          10.00        1.84      5751    1.37      1.000
tures of very evolved late-type giants and supergiants, such       RGB:tip          12.17       2730.      2602    256.      0.668
as α1 Her (see Schr¨der & Cuntz 2007, Fig. 4 in particular),
                     o                                             ZA-He            12.17        53.7      4667    11.2      0.668
and even later stages of stellar evolution (Dyck et al. 1996,      AGB:tip          12.30       2090.      3200    149.      0.546
and van Belle et al. 1996, 1997).                                  AGB:tip-TP       12.30       4170.      3467    179.      0.544
     The evolution model of the Sun presented here uses an
                                                                  (note: 1.00 AU = 215 R⊙ )
opacity grid that matches the empirical solar metallicity of
Anders & Grevesse (1989), Z = 0.0188, derived from atmo-
spheric models with simple 1D radiative transfer – an ap-         giant branch” in the HRD) – at first very gradually, but
proach consistent with our evolution models. Together with        then accelerating. At an age of 12.167 Gy, the Sun will have
X = 0.700 and Y = 0.2812, there is a good match with              reached the tip of the RGB, with a maximum luminosity of
present-day solar properties derived in the same way (see         2730 L⊙ .
Pols et al. 1995). We note that the use of 3D-hydrodynamic             In order to quantify the mass-loss rate of the evolved,
modelling of stellar atmospheres and their radiative transfer     cool solar giant at each time-step, we use the new and well-
may lead to a significantly lower solar abundance scale (e.g.,     calibrated mass-loss formula for ordinary cool winds (i.e.,
Asplund, Grevesse & Sauval 2005, who quote Z = 0.0122),                                         o
                                                                  not driven by dust) of Schr¨der & Cuntz (2005, 2007). This
but these lower values are still being debated, and create        relation is, essentially, an improved Reimers’ law, physically
some problems with helioseismology. Of course, using lower        motivated by a consideration of global chromospheric prop-
metallicities with an evolution code always results in more       erties and wind energy requirements:
compact and hotter stellar models. Hence, if we used a lower                                  3.5
                                                                  ˙     L∗ R∗ Teff                          g⊙
Z our code would plainly fail to reproduce the present-day        M = η                             1+                        (1)
                                                                         M∗ 4000 K                       4300 g∗
Sun, and the reliability of more evolved models with lower
Z must therefore also be seriously doubted.                       with η = 8×10−14 M⊙ y−1 , g⊙ = solar surface gravitational
     The resulting solar evolution model suggests an age of       acceleration, and L∗ , R∗ , and M∗ in solar units.
the present-day MS Sun of 4.58 Gy (±0.05 Gy), counted from                                                               o
                                                                       This relation was initially calibrated by Schr¨der &
its zero-age MS start model, which is well within the range       Cuntz (2005) with the total mass loss on the RGB, using
of commonly accepted values for the real age of the Sun and       the blue-end (i.e., the least massive) horizontal-branch (HB)
the solar system (e.g. Sackmann et al. 1993). Our model also      stars of globular clusters with different metallicities. This
confirms some well-established facts: (1) The MS-Sun has           method avoids the interfering problem of temporal mass-
already undergone significant changes, i.e., the present solar     loss variations found with individual giant stars and leaves
luminosity L exceeds the zero-age value by 0.30 L⊙ , and the      an uncertainty of the new η-value of only 15%, just under
zero-age solar radius R was 11% smaller than the present          the individual spread of RGB mass-loss required to explain
value. (2) There was an increase of effective temperature Teff      the width of HBs.
from, according to our model, 5596 K to 5774 K (±5 K). (3)                         o
                                                                       Later, Schr¨der & Cuntz (2007) tested their improved
The present Sun is increasing its average luminosity at a rate    mass-loss relation with six nearby galactic giants and su-
of 1% in every 110 million years, or 10% over the next billion    pergiants, in comparison with four other, frequently quoted
years. All this is completely consistent with established solar   mass-loss relations. All but one of the tested giants are AGB
models like the one of Gough (1981).                              stars, which have (very different) well-established physical
     Certainly, the solar MS-changes and their consequences       properties and empirical mass-loss rates, all by cool winds
for Earth are extremely slow, compared to the current cli-        not driven by radiation-pressure on dust. Despite the afore-
mate change driven by human factors. Nevertheless, solar          mentioned problem with the inherent time-variability of this
evolution will force global warming upon Earth already in         individual-star-approach, the new relation (equation (1))
the “near” MS future of the Sun, long before the Sun starts       was confirmed to give the best representation of the cool,
its evolution as a giant star (see our discussion of the hab-     but not “dust-driven” stellar mass-loss: it was the only one
itable zone in Section 3).                                        that agreed within the uncertainties (i.e., within a factor of
     At an age of 7.13 Gy, the Sun will have reached its high-    1.5 to 2) with the empirical mass-loss rates of all giants.
est Teff of 5820 K, at a luminosity of 1.26 L⊙ . From then on,     Hence, since the future Sun will not reach the critical lu-
the evolving MS Sun will gradually become cooler, but its         minosity required by a “dust-driven” wind (see Section 5),
luminosity will continue to increase. At an age of 10.0 Gy,       we here apply equation (1) to describe its AGB mass-loss as
the solar effective temperature will be back at Teff = 5751 K,      well as its RGB mass-loss.
while L = 1.84 L⊙ , and the solar radius then will be 37%              The exact mass-loss suffered by the future giant Sun
larger than today. Around that age, the evolution of the          has, of course, a general impact on the radius of the solar
Sun will speed up, since the solar core will change from cen-     giant, since the reduced gravity allows for an even larger
tral hydrogen-burning to hydrogen shell-burning and start         (and cooler) supergiant. The luminosity, however, is hardly
to contract. In response, the outer layers will expand, and       affected because it is mostly set by the conditions in the con-
the Sun will start climbing up the RGB (the “red” or “first        tracting core and the hydrogen-burning shell. In total, our
4               o
      K.-P. Schr¨der and R.C. Smith
solar evolution model yields a loss of 0.332 M⊙ by the time
the tip-RGB is reached (for η = 8 × 10−14 M⊙ y −1 ). This is
a little more than the 0.275 M⊙ obtained by Sackmann et
al. (1993), who used a mass-loss prescription based on the
original, simple Reimers’ relation. Furthermore, our evolu-
tion model predicts that at the very tip of the RGB, the
Sun should reach R = 256 R⊙ = 1.2 AU (see Fig. 1), with
L = 2730L⊙ and Teff = 2602 K. More details are given in
Table 1.
     By comparison, a prescription of the (average) RGB
mass-loss rate with η = 7 × 10−14 M⊙ y−1 , near the lower
error limit of the mass-loss calibration with HB stars, yields
a solar model at the very tip of the RGB with R = 249R⊙ ,
L = 2742L⊙ , Teff = 2650 K, and a total mass lost on the
RGB of 0.268M⊙ . With η = 9 × 10−14 M⊙ y−1 , on the other
hand, the Sun would reach the very tip of the RGB with
R = 256R⊙ , L = 2714L⊙ , Teff = 2605 K, and will have lost          Figure 1. Solar radius evolution during the RGB and AGB
                                                                   phases. Included for comparison (dashed curve) is the potential
a total of 0.388M⊙ . While these slightly different possible
                                                                   orbital radius of planet Earth, taking account of solar mass loss
outcomes of solar tip-RGB evolution – within the uncer-
                                                                   but neglecting any loss of orbital angular momentum. The labels
tainty of the mass-loss prescription – require further discus-     on the curve for the solar radius show the mass of the Sun in
sion, which we give in Section 4.3, the differences are too         units of its present-day mass.
small to be obvious on the scale of Fig. 1.
     With the reduced solar mass and, consequently, lower
gravitational attraction, all planetary orbits – that of the       prior mass loss from the giant Sun is essential for modelling
Earth included – are bound to expand. This is simply               this phase reliably.
a consequence of the conservation of angular momentum
ΛE = ME vE rE , while the orbital radius (i.e. rE ) adjusts
to a new balance between centrifugal force and the reduced         3   EVOLUTION OF THE HABITABLE ZONE
gravitational force of the Sun, caused by the reduced so-
                                                                   The Earth currently sits in the ‘habitable zone’ in the so-
lar mass MSun (t). Substituting vE = GMSun (t)/rE in ΛE
                                                                   lar system, that is, the region in which conditions on the
yields rE ∝ Λ2 /MSun (t). For this conservative case, we find
               E
                                                                   Earth – in particular the average planetary temperature –
that rE is 1.50 AU for the case η = 8 × 10−14 M⊙ y −1 . For
                                                                   are favourable for life. There are various precise definitions
the smaller (7 × 10−14 ) and larger (9 × 10−14 ) values of η,
                                                                   of ‘habitability’ in the literature, and a useful overview of
we find, respectively, rE = 1.37 AU and rE = 1.63 AU, so
                                                                   habitable zones in the wider context of extrasolar planetary
in all cases the orbital radius is comfortably more than the
                                                                   systems is given by Franck et al. (2002). For the current pa-
solar radius, when angular momentum is conserved.
                                                                   per, a convenient definition is that a planet is habitable if
     Section 4.1 provides a treatment of the more realistic
                                                                   the conditions on it allow the presence of liquid water on its
case, in which angular momentum is not conserved. We have
                                                                   surface. This may allow extremes of temperature that would
taken great care in determining the mass-loss and other pa-
                                                                   make life uncomfortable if not impossible for humans, but
rameters for our models, because the best possible models of
                                                                   the argument is that life of any kind (at least any kind we
the evolution of solar mass and radius through the tip-RGB
                                                                   know about at present) requires water at some stage in its
phase are required to provide reliable results.
                                                                   life cycle. We shall adopt that definition in this paper, but
     The significant solar RGB mass loss will also shape the
                                                                   note that even with that apparently simple definition it is
later solar AGB evolution. Compared with models without
                                                                   not straightforward to calculate the width of the habitable
mass loss, the AGB Sun will not become as large and lu-
                                                                   zone.
minous, and will be shorter-lived, because it lacks envelope
                                                                         It may be instructive to begin with a calculation of the
mass for the core and its burning shells to “eat” into. In fact,
                                                                   mean planetary temperature in terms of a spherical black
the solar tip-AGB radius (149 R⊙ ) will never reach that of
                                                                   body, by assuming that the planetary body absorbs the so-
the tip-RGB (see Fig. 1), and AGB thermal pulses are no
                                                                   lar flux intercepted by its (circular) cross-sectional area and
threat to any planet which would have survived the tip-
                                                                   re-emits it spherically symmetrically at a black body tem-
RGB. Our evolution code resolved only the two final and
                                                                   perature T . Then (cf. SSA) T is given by
most dramatic thermal pulses (cf. Section 5).
                                                                                                1/2
     The regular tip-AGB luminosity of 2090 L⊙ will not ex-                               R
                                                                   T   =    (1 − A)1/4                Teff
ceed the tip-RGB value, either. Hence, as will be discussed                              2D
in Section 5, the tip-AGB Sun will not develop a sustained                                        R
                                                                                                            1/2
                                                                                                                  1AU   1/2
dust-driven superwind but will stay short of the critical lu-          =    0.0682 (1 − A)1/4                                 Teff   (2)
                                                                                                  R⊙               2D
                                                    o
minosity required by dust-driven winds (see Schr¨der et al.
1999). The very tip of the AGB coincides with a thermal            where D is the distance of the body from the centre of the
pulse (TP), after which the giant briefly reaches a peak lu-        Sun, R is the radius of the Sun, A is the Bond albedo of the
minosity of 4170 L⊙ , but at a higher Teff = 3467 K than on         Earth and Teff is the effective temperature of the Sun. On
the RGB (see Table 1 and Section 5), keeping the radius            that basis, taking Teff = 5774 K and R = R⊙ (Table 1), and
down to 179 R⊙ . Again, the best possible treatment of all         A = 0.3 (Kandel & Viollier 2005), we find T (1 AU) = 255 K.
                                                                                Distant future of Sun and Earth                5
But the actual mean temperature of the Earth at present is        moment the short-timescale (decades to centuries) problems
33 K warmer, at T = 288 K. This demonstrates the warming          currently being introduced by climate change, we may ex-
effect of our atmosphere, which becomes significantly more          pect to have about one billion years of time before the solar
important with higher temperature (see below).                    flux has increased by the critical 10% mentioned earlier. At
      In fact, there are various complex, partly antagonistic     that point, neglecting the effects of solar irradiance changes
atmospheric feedback mechanisms (for example, the green-          on the cloud cover, the water vapour content of the atmo-
house effect, the variation of planetary albedo with the pres-     sphere will increase substantially and the oceans will start
ence of clouds, snow and ice, and the carbonate-silicate cycle    to evaporate (Kasting 1988). An initially moist greenhouse
which determines the amount of carbon dioxide in the at-          effect (Laughlin 2007) will cause runaway evaporation until
mosphere) that act to change the surface temperature from         the oceans have boiled dry. With so much water vapour in
what it would be in the absence of an atmosphere. These           the atmosphere, some of it will make its way into the strato-
mechanisms have been carefully discussed by Kasting, Whit-        sphere. There, solar UV will dissociate the water molecules
mire & Reynolds (1993), who conclude that a conservative          into OH and free atomic hydrogen, which will gradually es-
estimate of the current habitable zone (HZ) stretches from        cape, until most of the atmospheric water vapour has been
0.95 AU to 1.37 AU. We shall adopt their result for the lim-      lost. The subsequent dry greenhouse phase will raise the sur-
ited purposes of this paper. It can be adjusted in a simple-      face temperature significantly faster than would be expected
minded way to allow for the evolution of the Sun by scaling       from our very simple black-body assumption, and the ulti-
the inner and outer HZ radii rHZ,i , rHZ,o with the changing      mate fate of the Earth, if it survived at all as a separate
solar luminosity LSun (t): rHZ ∝ LSun (t). In this way, the       body (cf. Section 4), would be to become a molten remnant.
respective critical values of solar irradiance derived by Kast-
ing et al. (1993) for the inner and outer edge of the HZ are
maintained.                                                       4     THE INNER PLANETARY SYSTEM
      Certainly, with the 10% increase of solar luminosity over         DURING TIP-RGB EVOLUTION
the next 1 Gy (see previous section), it is clear that Earth
will come to leave the HZ already in about a billion years        After 12 Gy of slow solar evolution, the final ascent of the
time, since the inner (hot side) boundary will then cross         RGB will be relatively fast. The solar radius will sweep
1 AU. By the time the Sun comes to leave the main se-             through the inner planetary system within only 5 million
quence, around an age of 10 Gy (Table 1), our simple model        years, by which time the evolved solar giant will have
predicts that the HZ will have moved out to the range 1.29        reached the tip of the RGB and then entered its brief (130
to 1.86 AU. The Sun will have lost very little mass by that       million year) He-burning phase. The giant will first come to
time, so the Earth’s orbital radius will still be about 1 AU –    exceed the orbital size of Mercury, then Venus. By the time
left far behind by the HZ, which will instead be enveloping       it approaches Earth, the solar mass-loss rate will reach up to
the orbit of Mars.                                                2.5 × 10−7 M⊙ y−1 and lead to some orbital expansion (see
      By the time the Sun reaches the tip of the RGB, at          Section 2). But the extreme proximity of the orbiting planet
12.17 Gy, the Earth’s orbital radius will only have expanded      to the solar photosphere requires the consideration of two
to at most 1.5 AU, but the habitable zone will have a range       effects, which both lead to angular momentum loss and a
of 49.4 to 71.4 AU, reaching well into the Kuiper Belt! The       fatal decrease of the orbital radius of planet Earth.
positions of the HZ boundaries are not as well determined
as these numbers suggest, because in reality the scaling for
                                                                  4.1    Tidal interaction
the boundaries of the HZ almost certainly depends also on
how clouds are affected by changes in the solar irradiance.        For the highly evolved giant Sun, we may safely assume
These effects are complex and uncertain (cf. Kasting 1988),        (cf. Section 1) that it has essentially ceased to rotate, after
and may increase or decrease the speed at which the HZ            nearly 2 billion years of post-MS magnetic braking acting on
drifts outwards. But none the less it seems clear that the        the hugely expanded, cool RGB giant. Consequently, any
HZ will move out past the Earth long before the Sun has           tidal interaction with an orbiting object will result in its
expanded very much, even if the figure of one billion years        suffering a continuous drag by the slightly retarded tidal
is a rather rough estimate of how long we have before the         bulges of the giant solar photosphere.
Earth is uninhabitable.                                                As shown in Section 2, the orbital radius of planet Earth
      In other planetary systems around solar-type stars, con-    rE depends on the angular momentum squared, by the equa-
ditions may be different, and it may even be possible for life     tion
to start during a star’s post-main-sequence evolution, if a
                                                                              Λ2 (t)
                                                                               E
planet exists at a suitable distance from the star. This pos-     rE =    2
                                                                                         .                                   (3)
                                                                         ME   G MSun (t)
sibility is discussed by Lopez, Schneider & Danchi (2005),
who also give a general discussion of the evolution of habit-     Hence, the terrestrial orbit reacts quite sensitively to any
able zones with time. However, they use the evolution mod-        loss of angular momentum, by shrinking.
els of Maeder & Meynet (1988), which do not agree as well              The retardation of the tidal bulges of the solar photo-
as ours with the colours and observed Teff ’s of the red gi-       sphere will be caused by tidal friction in the outer convec-
ants in star clusters (see, e.g., illustrations given by Meynet   tive envelope of the RGB Sun. This physical process was
et al. 1993), and which predict a very different behaviour for     analyzed, solved and applied by J.-P. Zahn (1977, 1989, and
the solar radius; so their results are not directly comparable    other work referred to therein), and successfully tested with
with ours.                                                        the synchronization and circularization of binary star orbits
      What will happen on the Earth itself? Ignoring for the      by Verbunt & Phinney (1995). In a convective envelope, the
6               o
      K.-P. Schr¨der and R.C. Smith
main contribution to tidal friction comes from the retarda-                In the case of supersonic motion (with a Mach num-
tion of the equilibrium tide by interaction with convective          ber1 of the order of 2 to 3) in a gaseous medium, dynamical
motions. For a circular orbit, the resulting torque Γ exerted        friction consists in about equal shares of the collisionless,
on planet Earth by the retarded solar tidal bulges is given          gravitational interaction with its wake and of the friction it-
by (Zahn 1977; Zahn 1989, Eq.11):                                    self. In her study, Ostriker (1999, Fig. 3) finds that the drag
                                6                                    force exerted on the object in motion is
      λ2 2       2   RSun
Γ=6      q MSun RSun                (Ω − ω).                  (4)
      tf              rE
                                                                     Fd = λd 4πρ (GME /cs )2                                      (5)
Here, the angular velocity of the solar rotation is sup-
posed to be Ω = 0, while that of the orbiting Earth,                 where λd is of the order of 1 to 3. The numerical simulations
                                 3
ω(t) = 2π/PE (t) = Λ−3 (t)ME (GMSun (t))2 , will vary both                     a
                                                                     made by S´nchez-Salcedo & Brandenburg (2001) are in gen-
with the decreasing angular momentum Λ(t) (= 2.67 ×                  eral agreement with the results of Ostriker (1999). Here cs
1040 kg m2 s−1 at present) and with the solar mass in the            is the speed of sound, which in a stellar chromosphere is
final solar RGB stages. The exerted torque scales with                about 8 km s−1 , and ρ is the gas density (SI units). The lat-
the square of the (slowly increasing) mass ratio q(t) =              ter quantity is the largest source of uncertainty, as we can
ME /MSun (t) (= 3.005 × 10−6 at present), because q                  only make guesses (see below) as to what the gas density
determines the magnitude of the tidal bulges. tf (t) =               in the lower giant solar chromosphere will be. The angular
            2
(MSun (t)RSun (t)/LSun (t))1/3 ≈ O(1 y) is the convective fric-      momentum loss resulting from this drag is simply
tion time (Zahn 1989, Eq.7), and the coefficient λ2 depends
on the properties of the convective envelope. For a fully con-       dΛ/dt = −Fd rE ,                                             (6)
vective envelope (Zahn 1989, Eq.15), with a tidal period ≈
O(1 y), comparable to 2tf , we may use λ2 ≈ 0.019 α4/3 ≈             and the corresponding life-time of the orbital angular mo-
0.038 (with a convection parameter of our tip-RGB solar              mentum is τ = Λ/ | dΛ/dt |, as above.
model of α ≈ 1.7). This coefficient appears to be the main                  For the lower chromosphere of the K supergiant ζ Aur,
source of uncertainty (see Section 4.3), because it is related       employing an analysis of the additional line absorption in
to the simplifications of the mixing length theory (MLT).             the spectrum of a hot companion in chromospheric eclipse,
     With the properties of the tip-RGB Sun, a typical value              o
                                                                     Schr¨der, Griffin & Griffin (1990) found an average hydrogen
of the tidal drag acting on planet Earth is Γ = dΛ/dt =              particle density of 7 × 1011 cm−3 at a height of 2 × 106 km.
−3.3 × 1026 kg m2 s−2 , which gives a typical orbital angular        Alternatively, we may simply assume that the density of the
momentum decay time of τ =| Λ/Γ |= 2.6 × 106 y. This is              lower solar chromosphere scales with gravity g, which will be
comparable to the time spent by the Sun near the tip-RGB;            lower by 4.7 orders of magnitude on the tip-RGB, while the
since a loss of only ≈ 10% of the angular momentum will              density scale-height scales with g −1 (as observations of cool
be sufficient to reduce the orbital radius (by 20%) to lower                                                              o
                                                                     giant chromospheres seem to indicate, see Schr¨der 1990).
it into the solar giant photosphere, this order-of-magnitude         The chromospheric models of both Lemaire et al. (1981) and
calculation illustrates clearly that tidal interaction is crucial.   Maltby et al. (1986) suggest particle densities of the order of
Its full consideration requires a timestep-by-timestep com-          1017 cm−3 at a height of 100 km, and a scale height of that
putation of the loss of orbital angular momentum; at each            order for the present, low solar chromosphere. Scaled to tip-
time-step of the solar evolution calculation, we use equa-           RGB gravity, that would correspond to a particle density
tion (4), together with the radii and masses of our solar            of 2 × 1012 cm−3 , or ρ ≈ 4 × 10−9 kg m−3 , at a height of
evolution model, to compute the change in angular momen-             5 × 106 km (0.03 AU), and a density scale height of that
tum, and then use equation (3) to compute the change in              same value.
the orbital radius, and hence the new orbital period of the               For the computation of the orbital angular momentum
Earth. Section 4.3 presents the result, which also takes into        loss of the Earth, presented below (see Figures 2 and 3), we
account the relatively small additional angular momentum             apply the latter, rather higher values of the future chromo-
losses by dynamical drag, as discussed in the next section.          spheric gas density, together with the (also more pessimistic)
                                                                     assumption of λd = 3 (using cs = 8 km s−1 ). The typical an-
                                                                     gular momentum decay-time by dynamical friction in the
4.2   Dynamical friction in the lower chromosphere                   low (h ≈ 0.03 AU) chromosphere of the tip-RGB solar giant
A further source of angular momentum loss by drag is dy-             is 14 million years – significantly longer than that for tidal
namical friction, from which any object suffers in a fairly           interaction. Hence, this illustrates that dynamical friction is
close orbit, by its supersonic motion through the gas of             of interest only in the lowest chromospheric layers, adding
the then very extended, cool solar giant chromosphere. In a          there just a little to the drag exerted by tidal interaction.
different context, dynamical drag exerted by a giant atmo-            None the less, we include it, using equations (5) and (6) to
sphere has already been considered by Livio & Soker (1984).          calculate the additional angular momentum change to be
But the specific problem here is to find an adequate de-               included in equation (3).
scription of the density structure of the future cool solar gi-
ant. Fortunately, as it turns out (see below), dynamical drag
                         o
will play only a minor rˆle, very near the solar giant photo-
sphere, and the total angular momentum loss is dominated
by the tidal interaction described above. An approximate             1 Note that vE ∝ MSun (t), and so the Mach number is somewhat
treatment of the drag is therefore adequate, and we use the          lower than would be expected from the present orbital velocity of
recent study by Ostriker (1999).                                     the Earth of about 30 km s−1 .
                                                                                  Distant future of Sun and Earth                7
                                                                     ignition and not reach the horizontal branch at all. And the
                                                                     full width of the HB towards lower Teff is achieved already
                                                                     with an η of 7 × 10−14 M⊙ y−1 . Furthermore, the benefit of
                                                                     larger orbits with a reduced solar mass is to some extent
                                                                     compensated for by a larger solar giant.
                                                                           Dynamical drag does not become important until the
                                                                     planet is already very near the photosphere, i.e., after tidal
                                                                     drag has already lowered the orbit. Hence, the most signif-
                                                                     icant uncertainty here comes from the scaling of the tidal
                                                                     friction coefficient λ2 (of Zahn, 1989). For this reason, we
                                                                     computed several alternative cases, and from these we find:
                                                                           (1) With the mass-loss rate unchanged, the value of λ2
                                                                     would have to be significantly smaller for an escape from
                                                                     the “doomsday” scenario, i.e., less than 0.013, instead of
                                                                     our adopted value of 0.038. But Zahn’s scaling of λ2 has
                                                                     been empirically confirmed within a factor of 2, if not bet-
Figure 2. The final 4 million years of solar evolution before the     ter (see Verbunt & Phinney, 1995). Very recently, realistic
tip-RGB, showing the radii of the Sun and of the orbit of planet     3D simulations of the solar convection have also resulted in
Earth (dashed curve) – taking account of angular momentum
                                                                     an effective viscosity which matches that of Zahn’s prescrip-
losses by tidal interaction and supersonic drag. The labels on the
solar radius track give values of MSun (t)/M⊙ , as in Figure 1.
                                                                     tion surprisingly well (Penev et al. 2007). And Rybicki &
                                                                     Denis (2001), by comparison, used a value (K2 = 0.05 in
                                                                     the notation of their very similar calculation of tidal angu-
                                                                     lar momentum loss) which is entirely consistent with Zahn’s
4.3   “Doomsday” confirmed                                            scaling of λ2 for a convection parameter of α = 2.
As explained in the previous two sections, we use equations                (2) We then considered solar evolution models with a
(3) to (6) to compute, at each time-step of our evolutionary         reasonably larger mass-loss rate (η = 9 × 10−14 M⊙ y−1 ) in
calculation, a detailed description of the orbital evolution for     combination with tidal friction coefficients of 1/1, 2/3 and
planet Earth in the critical tip-RGB phase of the Sun under          1/2 of the one given by Zahn. In each of these cases, planet
the influence of tidal interaction and dynamical drag. The            Earth would not be able to escape doomsday but would
resulting evolution both of the orbital radius of the Earth          face a delayed engulfment by the supergiant Sun – 470,000,
and of the radius of the solar giant is shown in Fig. 2. This        230,000 and 80,000 years before the tip-RGB is reached,
shows that, despite the reduced gravity from a less massive          respectively.
tip-RGB Sun, the orbit of the Earth will hardly ever come to               (3) Finally, we checked the outcome for a reasonably
exceed 1 AU by a significant amount. The potential orbital            lower mass-loss rate (η = 7 × 10−14 M⊙ y−1 ) in combination
growth given by the reduced solar mass is mostly balanced            with the same tidal friction coefficients as above. The en-
and, eventually, overcome by the effects of tidal interaction.        gulfment would then happen rather earlier than with more
Near the very end, supersonic drag also becomes a significant         mass-loss – 540,000, 380,000 and 270,000 years before the
source of angular momentum loss.                                     tip-AGB is reached.
     As shown by Fig. 2, engulfment and loss of planet Earth               These computations confirm that reducing the solar
will take place just before the Sun reaches the tip of the           mass enlarges the planetary orbit more than the tip-RGB
RGB, 7.59 Gy (±0.05 Gy) from now. According to our calcu-            solar radius, so that the best way to avoid the doomsday
lation, it occurs when the RGB Sun has still another 0.25 AU         scenario would be to have as high a mass-loss rate as possi-
to grow, about 500,000 years before the tip-RGB. Of course,          ble. However, we believe that the value of η in case (2) above
Mercury and Venus will already have suffered the same fate            already is as high as it can be without violating agreement
as Earth some time before – respectively, 3.8 and 1.0 million        of evolved models with observations, and that the smallest
years earlier.                                                       value used there for the tidal friction coefficient is also at
     As mentioned in the introduction, a similar calculation         the limits of what is allowed by observational constraints.
was already carried out in the context of extra-solar plan-          The only possible escape would be if our solar giant models
ets by Rasio et al. (1996), who basically came to the same           were too cool (by over 100 K in case 2), and therefore larger
conclusions; their fig. 2 is reminiscent of ours. They also em-       than the real Sun will be. Hence, to avoid engulfment by the
ployed the orbital decay rate predicted by Zahn’s theory, but        tip-RGB Sun would require that all three parameters (η, λ2
their solar evolution model used the old Reimers mass-loss           and Teff ) were at one edge of their uncertainty range, which
relation, and they did not make any adjustments to match             seems improbable. Rather, our computations confirm, with
the effective temperatures found empirically at the tip of the        reasonable certainty, the classical “doomsday” scenario.
giant branches (see Section 2).
     Do the remaining uncertainties allow the possibility for
                                                                     4.4   “Doomsday” avoidable?
Earth to escape the “doomsday” scenario? As far as the
mass-loss alone is concerned, this seems unlikely: according         Even though this is an academic question, given the hos-
to the study of HB stars in globular clusters by Schr¨der  o         tile conditions on the surface of a planet just missing this
& Cuntz (2005), η is remarkably well constrained and can-            “doomsday”scenario, we may ask: what is the minimum ini-
not exceed 9 × 10−14 M⊙ y−1 , or the total RGB mass-loss             tial orbital radius of a planet in order for it to “survive”?
would become so large that the tip-RGB star would miss He-           Fig. 3 shows, by the same computation as carried out for Fig.
8               o
      K.-P. Schr¨der and R.C. Smith




Figure 3. As Fig. 2, but for a planet with a present orbital radius   Figure 4. Solar mass loss during the final 1 million years on the
of 1.15 AU.                                                           AGB will remain mainly of the order of 2 × 10−7 M⊙ y−1 and not
                                                                      provide sufficient CS shell mass to form a regular PN. Only the
                                                                      last two TP’s (tip-AGB and post-AGB, see text) are resolved.
2, that an initial orbital radius of 1.15 AU is sufficient for
any planet to pass the tip-RGB of a star with Mi = 1.0 M⊙ .
Since, as shown in Section 5, the tip-AGB Sun will not reach          that could move slowly outwards to maintain habitable con-
any similarly large extent again, such a planet will eventu-          ditions would, on cost and energy grounds, necessarily be
ally be orbiting a White Dwarf.                                       confined to a small fraction of the human population – with
      A more general discussion of planetary survival during          all the political problems that that would produce – plus
post-main-sequence evolution has been given by Villaver &             perhaps a tiny proportion of other species. None the less,
Livio (2007), who suggest that an initial distance of at least        the asteroidal fly-by scheme has its own problems, not least
3 AU is needed for the survival of a terrestrial-size planet          the danger of a benign close approach turning into a catas-
when one also takes into account the possible evaporation             trophic accidental collision, and possibly also triggering or-
of the planet by stellar heating. However, they use stellar           bital instability – cf. also Debes & Sigurdsson (2002).
models and mass-loss rates that have the maximum radius
and mass loss occurring on the AGB. That has been the
expected result for many years, but is quite different from
                                                                      5   TIP-AGB SOLAR EVOLUTION
what we find (Section 5 and Table 1) with the improved
                                    o
mass-loss formulation of Schr¨der & Cuntz (2005, 2007).               The loss of 1/3 of the solar mass during the rise to the tip of
Hence, Villaver & Livio’s results may be unduly pessimistic.          the RGB will make a significant impact on the further evolu-
      In any case, it is clear that terrestrial planets can survive   tion as an AGB star. There is very little shell mass left, into
if sufficiently far from their parent star. If it were possible         which the two burning shells (H, followed by He) can ad-
to increase the orbital radius from its initial value, then an        vance (on a radial mass scale). Hence, the C/O core cannot
increase of only 8% of angular momentum should yield the              grow as much as with a conservative model without mass
pre-RGB orbital size required by planet Earth to escape               loss, and the whole core region will not contract as much,
engulfment. Is that conceivable?                                      either. Consequently, the AGB luminosity, determined by
      An ingenious scheme for doing so which, in the first             the density and temperature in the H-burning shell, will not
place, could increase the time-scale for habitation by in-            reach as high levels as in a conservative AGB model, and nei-
telligent life for the whole of the Sun’s MS life-time, was           ther will the AGB radius of the late future Sun (see Table
proposed by Korycansky, Laughlin & Adams (2001). They                 1).
pointed out that a suitable encounter of the Earth every                   According to our evolution model, the regular tip-AGB
6000 years or so with a body of large asteroidal mass could           evolution will be ended with a luminosity of only 2090 L⊙ ,
be arranged to move the orbit of the Earth outwards; Kuiper           Teff = 3200 K, and R = 149 R⊙ . The AGB mass-loss rate,
Belt objects might be the most suitable. The energy require-                                              o
                                                                      according to the relation of Schr¨der & Cuntz (2005), will
ments could be reduced by incorporating additional encoun-            reach only 2.0 × 10−7 M⊙ y−1 (see Fig. 4), since the lumi-
ters with Jupiter and/or Saturn. Although still very large            nosity will not be sufficient to drive a dust-driven wind (see
by today’s standards, the energy requirements remain small                 o
                                                                      Schr¨der et al. 1999). Also, even if it did: only a little shell
compared to those for interstellar travel.                            mass will have been left to lose after the RGB phase, only
      On the face of it, this scheme seems far-fetched, but           0.116M⊙ .
Korycansky et al. (2001) show that it is in principle pos-                 Hence, for this non-dust-driven AGB solar mass-loss,
sible, both technically and energetically, although currently         we have adopted the same mass-loss description as above
somewhat beyond our technical capabilities; however, there            (equation (1)). This mass-loss, in combination with our solar
is no immediate hurry to implement the scheme, which could            evolution model, yields the following prediction: during the
await the development of the relevant technology. It would            final 30,000 y on the very tip-AGB, which are crucial for any
have the advantage of improving conditions for the whole              build-up of sufficient CS (circumstellar) material to form
biosphere, whereas any scheme for interplanetary ‘life rafts’         a PN, the solar giant will lose only 0.006 M⊙ . A further
                                                                               Distant future of Sun and Earth                   9
0.0015 M⊙ will be lost in just 1300 years right after a final      ACKNOWLEDGMENTS
thermal pulse (TP) on the tip-AGB. That marks the very
                                                                  KPS is grateful for travel support received from the As-
end of AGB evolution, and it allows the solar supergiant
                                                                  tronomy Centre at Sussex through a PPARC grant, which
briefly to reach a luminosity of 4170 L⊙ and R = 179 R⊙ ,
                                                                  enabled the authors to initiate this research project in the
with a mass-loss rate of 10−6 M⊙ y−1 , but with Teff already
                                                                  summer of 2006. We further wish to thank Jean-Paul Zahn
increased to 3467 K. Again, there will be no involvement
                                                                  for very helpful comments on his treatment of tidal friction
of a dust-driven wind. Since common PNe and their dusty
                                                                  and Adam Scaife of the Met Office’s Hadley Centre for sug-
CS envelopes reveal a dust-driven mass-loss history of more
                                                                  gesting changes to Sections 1 and 3.
like 10−5 to 10−4 M⊙ y−1 during the final 30,000 years of
tip-AGB evolution, we must conclude that the Sun will not
form such a PN.
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Description: Distant Future of the Earth and Sun Revisited, Schroeder and Smith, astronomy, astrophysics, cosmology, general relativity, quantum mechanics, physics, university degree, lecture notes, physical sciences